Trotter errors from dynamical structural instabilities of Floquet maps in quantum simulation

Abstract

We study the behavior of errors in the quantum simulation of spin systems with long-range multi-body interactions resulting from the Trotter-Suzuki decomposition of the time-evolution operator. We identify a regime where the Floquet operator underlying the Trotter decomposition undergoes sharp changes even for small variations in the simulation step size. This results in a time evolution operator that is very different from the dynamics generated by the targeted Hamiltonian, which leads to a proliferation of errors in the quantum simulation. These regions of sharp change in the Floquet operator, referred to as structural instability regions, appear typically at intermediate Trotter step sizes and in the weakly-interacting regime, and are thus complementary to recently revealed quantum chaotic regimes of the Trotterized evolution (Sieberer et al., npj Quantum Information 5, 1 (2019)). We characterize these structural instability regimes in $p$-spin models, transverse-field Ising models with all-to-all $p$-body interactions, and analytically predict their occurrence based on unitary perturbation theory. We further show that the effective Hamiltonian associated with the Trotter decomposition of the unitary time-evolution operator, when the Trotter-step size is chosen to be in the structural instability region, is very different from the target Hamiltonian, which explains the large errors that can occur in the simulation in the regions of instability. These results have implications for the reliability of near-term gate-based quantum simulators, and reveal an important interplay between errors and the physical properties of the system being simulated.